K-Ion Slides in Prussian Blue Analogues

We study the phenomenology of cooperative off-centering of K+ ions in potassiated Prussian blue analogues (PBAs). The principal distortion mechanism by which this off-centering occurs is termed a “K-ion slide”, and its origin is shown to lie in the interaction between local electrostatic dipoles that couple through a combination of electrostatics and elastic strain. Using synchrotron powder X-ray diffraction measurements, we determine the crystal structures of a range of low-vacancy K2M[Fe(CN)6] PBAs (M = Ni, Co, Fe, Mn, Cd) and establish an empirical link between composition, temperature, and slide-distortion magnitude. Our results reflect the common underlying physics responsible for K-ion slides and their evolution with temperature and composition. Monte Carlo simulations driven by a simple model of dipolar interactions and strain coupling reproduce the general features of the experimental phase behavior. We discuss the implications of our study for optimizing the performance of PBA K-ion battery cathode materials and also its relevance to distortions in other, conceptually related, hybrid perovskites.

1 Group Theory Table S1: Distortion modes for and corresponding distortion magnitudes at room temperature for K 2 Mn[Fe(CN) 6 ] relative to the unit cell given in Table S3.Primary distortions as discussed in the main text are highlighted in blue; the remaining distortions are secondary.Note that some irrep labels differ from those discussed in the main text as the parent structure has F m 3m symmetry and incorporates B-site order.

Measurements
Our powder X-ray diffraction data made use of the synchrotron X-ray source on the I11 beamline at Diamond Light Source, UK.The diffraction patterns of the sample were collected in capillary transmission geometry.A room temperature X-ray diffraction pattern was collected using the Mythen2 Position Sensitive Detector (PSD), two data collections of 20 seconds each were taken at angles 0.25 degrees apart, then summed to account for gaps in the detector coverage.
Measurements between 300 and 1000 K were performed using an FMB Oxford cyberstar hot air blower aimed side-on to the sample at the beam position.Data were collected by continually warming the sample at a rate of 6 K min −1 while collecting two 5 second scans with the PSD.

Ambient-temperature data: refinement details
The structural refinement for room temperature K 2 M[Fe(CN) 6 ] M = Cd, Mn, Fe, Co, Ni) was performed using a Rietveld refinement in TOPAS software, S2 in conjunction with distortion modes obtained from ISODISTORT.S3, S4 The parent cell was a cubic crystal structure of K 2 M[Fe(CN) 6 ] in F m 3m taken from the undistorted cubic cell from Ref. S5, with crystallographic details given in Table S3.K occupancies were allowed to refine freely and refinements returned an occupancy of at least 1.95 for all samples.
The uncertainties reported from the refinement in TOPAS are likely overly precise, hence why ICP-MS was employed to back-up the evidence.As further support for the sensitivity of the refinement to K occupancy a fit is included in Fig. S2 where the K occupancy was fixed to K 1.6 Mn[Fe(CN) 6 ].
Positions of all of the atoms were allowed to refine as a function of the 21 distortion modes generated by a distortion from F m 3m to P 2 1 /n, many of which were minor in contribution, but allowed by symmetry.In order to reduce the number of free parameters, the thermal displacement parameters for transition metals M and Fe were constrained to be equal; so too for those of the C and N sites.The K site was allowed to refine with an independent thermal displacement parameter.Finally, an anisotropic peakshape model for an monoclinic unit cell was applied to account for the different strains in different directions that individual crystallites will have, giving a better interpretation of the peak intensities.S6

Figure S1 :
Figure S1: Scanning electron micrographs of samples K2M[Fe(CN)6] with corresponding label for M and estimate of particle size.The variation in particle size scales somewhat with peak width in the XRD [Fig.3(b)].Each sample has corresponding two micrographs.The scale bar on each micrograph is 1 µ m.Domains in each crystal are clear from the aggregation of crystallites.

Figure S2 :
Figure S2: Rietveld fit to the X-ray powder diffraction pattern measured for K2Mn[Fe(CN)6] at room temperature, with fixed K occupancy to give the formula K1.6Mn[Fe(CN)6].Raw data in black, fit in red, and corresponding difference curve in grey (data -fit) is offset below the data.Tick marks denote reflection positions.

Figure S3 :
Figure S3: Rietveld fit to the X-ray powder diffraction pattern measured for K2Ni[Fe(CN)6] at room temperature.Raw data in black, fit in red, and corresponding difference curve in grey (data -fit) is offset below the data.Tick marks denote reflection positions.

Figure S4 :
Figure S4: Rietveld fit to the X-ray powder diffraction pattern measured for K2Co[Fe(CN)6] at room temperature.Raw data in black, fit in red, and corresponding difference curve in grey (data -fit) is offset below the data.Tick marks denote reflection positions.

Figure S5 :
Figure S5: Rietveld fit to the X-ray powder diffraction pattern measured for K2Fe[Fe(CN)6] at room temperature.Raw data in black, fit in red, and corresponding difference curve in grey (data -fit) is offset below the data.Tick marks denote reflection positions.

Figure S6 :
Figure S6: Rietveld fit to the X-ray powder diffraction pattern measured for K2Cd[Fe(CN)6] at room temperature.Raw data in black, fit in red, and corresponding difference curve in grey (data -fit) is offset below the data.Tick marks denote reflection positions.

Figure S7 :
Figure S7: Map of phase fractions for K2Mn[Fe(CN)6] between 790 and 1000 K. Monoclinic phase in orange, T ′ in black and T in teal.The anisotropic thermal displacement parameters for each tetragonal phase are plotted on the secondary axis with fits to refined regions illustrated.

Figure S8 :
Figure S8: Rietveld fit to the X-ray powder diffraction pattern measured for K2Mn[Fe(CN)6] at 803K.Raw data in black, fit in red, and corresponding difference curve in grey (data -fit) is offset below the data.Tick marks denote reflection positions for the monoclinic (orange) and T ′ (black) phases.

Figure S9 :
Figure S9: Rietveld fit to the X-ray powder diffraction pattern measured for K 2 Mn[Fe(CN) 6 ] at 898K.Raw data in black, fit in red, and corresponding difference curve in grey (data -fit) is offset below the data.Tick marks denote reflection positions for the T ′ (black) and T (teal) phases.
All samples were tested for elemental composition by ICP-MS.The main use of this technique is to measure the ratio of the transition metal to estimate vacancy content, where K + concentration can be inferred by charge balancing.This technique is notoriously unreliable in determining potassium content, S1 but it is supported by evidence from the Rietveld refinement of the room temperature XRD patterns.

Table S3 :
Crystallographic details for the F m 3m parent cubic structure.S5

Table S4 :
Crystallographic parameters for P 21/n structure of K 1.986(2) Ni[Fe(CN) 6 ] at room temperature from the fit the data in Fig. S3.

Table S5 :
Crystallographic parameters for P 21/n structure of K 1.9708(16) Co[Fe(CN) 6 ] at room temperature from the fit the data in Fig.S4.

Table S6 :
Crystallographic parameters for P 21/n structure of K 1.946(2) Fe[Fe(CN) 6 ] at room temperature from the fit the data in Fig.S5

Table S7 :
Crystallographic parameters for P 21/n structure of K 1.9736(18) Mn[Fe(CN) 6 ] at room temperature from the fit the data in Fig.4of the main text.

Table S8 :
Crystallographic parameters for P 21/n structure of K 1.954(2) Cd[Fe(CN) 6 ] at room temperature from the fit the data in Fig. S6.